Probabilistic Proof Research Articles

Probabilistic method to fundamental gap problems on the sphere

We provide a probabilistic proof of the fundamental gap estimate for Schrödinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo [Potential Anal. 44 (2016), pp. 423–442] for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei [J. Differential Geom. 112 (2019), pp. 347–389], as well as by C. He, G. Wei, and Qi S. Zhang [Amer. J. Math. 142 (2020), pp. 1161–1191]. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds.

Dual process in the two-parameter Poisson–Dirichlet diffusion

The two-parameter Poisson–Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson–Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman’s coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson–Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman’s Pólya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.

ESTARK: extending STARKs with arguments

STARK is a widely used transparent proof system that uses low-degree tests for proving the correctness of a computer program. STARK consumes an intermediate representation known as AIR that is more appropriate for programs with a relatively short and structured description. However, an AIR is not able to succinctly express non-equality constraints, leading to the incorporation of unwanted polynomials. We present the eSTARK protocol, a new probabilistic proof that generalizes the STARK family through the introduction of a more generic intermediate representation called eAIR. We describe eSTARK in the polynomial IOP model, which combines the optimized version of the STARK protocol with the incorporation of three arguments into the protocol. We also explain various techniques that enhance the vanilla STARK complexity, including optimizations applied to polynomial computations, and analyze the tradeoffs between controlling the constraint degree either at the representation of the AIR or inside the eSTARK itself.

A probabilistic proof of some integral formulas involving incomplete gamma functions

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ∫ 0 ∞ x − 2 ν cos ⁡ ( b x ) γ ( ν , α x 2 ) d x (for ν > 1 / 2 , b > 0, α > 0) and ∫ 0 ∞ x 2 ν − 1 cos ⁡ ( b x ) Γ ( − ν , α x 2 ) d x (for ν > 0, b > 0, α > 0), where γ(a, x) and Γ(a, x) are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas.

Microscopic derivation of Vlasov–Dirac–Benney equation with short-range pair potentials

We present a probabilistic proof of the mean-field limit and propagation of chaos of a N-particle system in three dimensions with pair potentials of the form N3β−1ϕ(Nβx) for β∈[0,17] and ϕ∈ L∞(R3)∩L1(R3) . In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov–Dirac–Benney system with delta-like interactions. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov–Dirac–Benney equation from the microscopic N-particle dynamics with a strong short range force.

Two-weight dyadic Hardy inequalities

We present various results concerning the two-weight Hardy inequality on infinite trees. Our main aim is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization in terms of a reverse Hölder inequality for trace measures, and one based on the well-known Muckenhoupt–Wheeden–Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so-called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so-called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy inequality, characterize the compactness of the Hardy operator, provide a list of open problems, and suggest some possible lines of future research.

Liouville theorem for [formula omitted]-harmonic maps under non-negative [formula omitted]-Ricci curvature for non-positive [formula omitted

Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan–Hadamard manifolds, which extends Cheng’s Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).

Two Comments on the Binomial Theorem

Summary We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course.

Polynomial approximations in a generalized Nyman–Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

Elliptic Harnack inequality for ℤd

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.

Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function $$f :L^m \rightarrow {\mathbb {F}}_q$$ , where $$L\subset {\mathbb {F}}_q$$ , a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed–Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed–Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed–Solomon codes of [Ben-Sasson et al., ICALP 2018] named “FRI protocol”. Compared to the FRI protocol, our IOPP for tensor products of Reed–Solomon codes achieves the same efficiency parameters. As for Reed–Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed–Muller codes compares well, all things equal.

Branching Brownian motion in a periodic environment and uniqueness of pulsating traveling waves

AbstractUsing one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating traveling waves of the Fisher–Kolmogorov–Petrovskii–Piskounov (F-KPP) equation in a periodic environment. This paper is a sequel to ‘Branching Brownian motion in a periodic environment and existence of pulsating travelling waves’ (Ren et al., 2022), in which we proved the existence of the pulsating traveling waves in the supercritical and critical cases, using the limits of the additive and derivative martingales of branching Brownian motion in a periodic environment.

A Probabilistic Proof of an Asymptotic Formula for the Modified Bessel Function

A Probabilistic Proof of an Asymptotic Formula for the Modified Bessel Function

A Collection of Results Relating the Geometry of Plane Domains and the Exit Time of Planar Brownian Motion

We prove a number of results relating exit times of planar Brownian motion with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.

Long-term concentration of measure and cut-off

We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains. We apply our discrete-time inequality to the well-studied Bernoulli–Laplace model of diffusion, and give a probabilistic proof of cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We also extend the notion of cut-off to chains with an infinite state space, and illustrate this in a second example, of a two-host model of disease in continuous time. We give a third example, giving concentration results for the supermarket model, illustrating the full generality and power of our results.

A probabilistic approach to Hilbert transforms on free group von Neumann algebras

The paper contains a probabilistic proof of the L p L^p -boundedness of the Hilbert transform in the context of a free group von Neumann algebra V N ( F q ) VN(\mathbb {F}_q) . The argument rests on noncommutative version of good- λ \lambda inequalities and yields a tight L log ⁡ L L\log L order of the L p L^p norms as p → 1 + p\to 1^+ and p → ∞ p\to \infty .

Spatial Isolation Implies Zero Knowledge Even in a Quantum World

Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP , as long as one is willing to make a suitable physical assumption : if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP * model captures this setting. In this work, we study the following question: Does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP * . Our proof consists of constructing a zero knowledge interactive probabilistically checkable proof with a strong algebraic structure, and then lifting it to the MIP * model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory.

A probabilistic proof of the Perron–Frobenius theorem

AbstractThe Perron–Frobenius theorem plays an important role in many areas of management science and operations research. This article provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic representation of the Perron–Frobenius eigenvalue and eigenvectors in terms of the dynamics of a Markov chain. The proof recovers conditions in both the finite‐dimensional and infinite‐dimensional settings under which the Perron–Frobenius eigenvalue and eigenvectors have been shown to exist by other methods. In addition to providing new insights, the probabilistic representations that arise can be used to produce a Monte–Carlo algorithm for computing the Perron–Frobenius eigenvalue and eigenvectors that will be explored elsewhere.

Provably Safe Artificial General Intelligence via Interactive Proofs

Methods are currently lacking to prove artificial general intelligence (AGI) safety. An AGI ‘hard takeoff’ is possible, in which first generation AGI1 rapidly triggers a succession of more powerful AGIn that differ dramatically in their computational capabilities (AGIn << AGIn+1). No proof exists that AGI will benefit humans or of a sound value-alignment method. Numerous paths toward human extinction or subjugation have been identified. We suggest that probabilistic proof methods are the fundamental paradigm for proving safety and value-alignment between disparately powerful autonomous agents. Interactive proof systems (IPS) describe mathematical communication protocols wherein a Verifier queries a computationally more powerful Prover and reduces the probability of the Prover deceiving the Verifier to any specified low probability (e.g., 2−100). IPS procedures can test AGI behavior control systems that incorporate hard-coded ethics or value-learning methods. Mapping the axioms and transformation rules of a behavior control system to a finite set of prime numbers allows validation of ‘safe’ behavior via IPS number-theoretic methods. Many other representations are needed for proving various AGI properties. Multi-prover IPS, program-checking IPS, and probabilistically checkable proofs further extend the paradigm. In toto, IPS provides a way to reduce AGIn ↔ AGIn+1 interaction hazards to an acceptably low level.

Probabilistic interpretations of nonclassic Adomian polynomials

The Adomian decomposition method (ADM) is a powerful tool for solving numerous nonlinear functional equations and a large class of initial/boundary value problems. The main task in the application of ADM is the computation of Adomian polynomials (APs). In addition to classic APs, Rach’s nonclassic APs, called class I-IV polynomials, are used to solve a wide range of nonlinear functional equations. In this paper, we present probabilistic interpretations for the nonclassic APs, including class V polynomials studied by Duan. We derive the recurrence relations for the computation of nonclassic APs using our approach. Some numerical examples are discussed to show that the probabilistic approach to compute the nonclassic APs is attractive and is also simple. Finally, a probabilistic proof is given for the known fact that the class IV APs are the classic APs. The probabilistic approach offers an alternative method to the existing analytical or combinatorial approach.